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Some questions to be answered:

1. How do I identify segments and lines related to circles?

2. How do I use the properties of a tangent to a circle?

1. How do I identify segments and lines related to circles?

2. How do I use the properties of a tangent to a circle?

1.
Circles

Chapter 10

Chapter 10

2.
Essential Questions

How do I identify segments and lines

related to circles?

How do I use properties of a tangent to a

circle?

How do I identify segments and lines

related to circles?

How do I use properties of a tangent to a

circle?

3.
A circle is the set of all points in a plane that are

equidistant from a given point called the center of the

circle.

Radius – the distance from the center to a point on the

circle

Congruent circles – circles that have the same radius.

Diameter – the distance across the circle through its

center

equidistant from a given point called the center of the

circle.

Radius – the distance from the center to a point on the

circle

Congruent circles – circles that have the same radius.

Diameter – the distance across the circle through its

center

4.
Diagram of Important Terms

center

radius

P

diameter

name of circle: P

center

radius

P

diameter

name of circle: P

5.
Chord – a segment whose endpoints are points on the

circle.

B

A

AB is a chord

circle.

B

A

AB is a chord

6.
Secant – a line that intersects a circle in two points.

N

M

MN is a secant

N

M

MN is a secant

7.
Tangent – a line in the plane of a circle that intersects

the circle in exactly one point.

T

S

ST is a tangent

the circle in exactly one point.

T

S

ST is a tangent

8.
Example 1

Tell whether the line or segment is best described as a chord, a

secant, a tangent, a diameter, or a radius.

H

a. AH tangent

b. EI diameter

B E

C F

c. DF chord

I G

d. CE radius

A D

Tell whether the line or segment is best described as a chord, a

secant, a tangent, a diameter, or a radius.

H

a. AH tangent

b. EI diameter

B E

C F

c. DF chord

I G

d. CE radius

A D

9.
Tangent circles – coplanar circles that

intersect in one point

intersect in one point

10.
Concentric circles – coplanar circles that

have the same center.

have the same center.

11.
Common tangent – a line or segment that is

tangent to two coplanar circles

Common internal tangent – intersects the segment

that joins the centers of the two circles

Common external tangent – does not intersect the

segment that joins the centers of the two circles

common external tangent

common internal tangent

tangent to two coplanar circles

Common internal tangent – intersects the segment

that joins the centers of the two circles

Common external tangent – does not intersect the

segment that joins the centers of the two circles

common external tangent

common internal tangent

12.
Example 2

Tell whether the common tangents are internal or external.

a. b.

common internal tangents common external tangents

Tell whether the common tangents are internal or external.

a. b.

common internal tangents common external tangents

13.
More definitions

Interior of a circle – consists of the points

that are inside the circle

Exterior of a circle – consists of the points

that are outside the circle

Interior of a circle – consists of the points

that are inside the circle

Exterior of a circle – consists of the points

that are outside the circle

14.
Point of tangency – the point at which a tangent line

intersects the circle to which it is tangent

point of tangency

intersects the circle to which it is tangent

point of tangency

15.
Perpendicular Tangent Theorem

If a line is tangent to a circle, then it is perpendicular to

the radius drawn to the point of tangency.

l

P

Q

If l is tangent to Q at P, then l QP.

If a line is tangent to a circle, then it is perpendicular to

the radius drawn to the point of tangency.

l

P

Q

If l is tangent to Q at P, then l QP.

16.
Perpendicular Tangent Converse

In a plane, if a line is perpendicular to a radius of a circle

at its endpoint on the circle, then the line is tangent to

the circle.

l

P

Q

If l QP at P, then l is tangent to Q.

In a plane, if a line is perpendicular to a radius of a circle

at its endpoint on the circle, then the line is tangent to

the circle.

l

P

Q

If l QP at P, then l is tangent to Q.

17.
Right Triangles

Pythagorean Theorem

Radius is perpendicular to the

tangent. < E is a right angle

C

43

E

45

11

D

Pythagorean Theorem

Radius is perpendicular to the

tangent. < E is a right angle

C

43

E

45

11

D

18.
Example 3

C

Tell whether CE is tangent to D. 43

E

Use the converse of the Pythagorean 45

Theorem to see if the triangle is right. 11

D

11 + 43 ? 45

2 2 2

121 + 1849 ? 2025

1970 2025

CED is not right, so CE is not tangent to D.

C

Tell whether CE is tangent to D. 43

E

Use the converse of the Pythagorean 45

Theorem to see if the triangle is right. 11

D

11 + 43 ? 45

2 2 2

121 + 1849 ? 2025

1970 2025

CED is not right, so CE is not tangent to D.

19.
Congruent Tangent Segments Theorem

If two segments from the same exterior point are tangent

to a circle, then they are congruent.

R

P

S

T

If SR and ST are tangent to P, then SR ST.

If two segments from the same exterior point are tangent

to a circle, then they are congruent.

R

P

S

T

If SR and ST are tangent to P, then SR ST.

20.
Example 4

AB is tangent to C at B. D

AD is tangent to C at D. x2 + 2

Find the value of x. C A

11

AD = AB

B

x2 + 2 = 11

x2 = 9

x = 3

AB is tangent to C at B. D

AD is tangent to C at D. x2 + 2

Find the value of x. C A

11

AD = AB

B

x2 + 2 = 11

x2 = 9

x = 3

21.
Central angle – an angle whose vertex is the center of a circle.

central angle

central angle

22.
Minor arc – Part of a circle that measures less

than 180°

Major arc – Part of a circle that measures

between 180° and 360°.

Semicircle – An arc whose endpoints are the

endpoints of a diameter of the circle.

Note : major arcs and semicircles are named with

three points and minor arcs are named

with two points

than 180°

Major arc – Part of a circle that measures

between 180° and 360°.

Semicircle – An arc whose endpoints are the

endpoints of a diameter of the circle.

Note : major arcs and semicircles are named with

three points and minor arcs are named

with two points

23.
Diagram of Arcs

A

minor arc: AB

major arc: ABD

D B

C

semicircle: BAD

A

minor arc: AB

major arc: ABD

D B

C

semicircle: BAD

24.
Measure of a minor arc – the measure of its

central angle

Measure of a major arc – the difference between

360° and the measure of its associated minor

arc.

central angle

Measure of a major arc – the difference between

360° and the measure of its associated minor

arc.

25.
Arc Addition Postulate

The measure of an arc formed by two adjacent arcs is the sum of

the measures of the two arcs.

A

C

mABC = mAB + mBC

B

The measure of an arc formed by two adjacent arcs is the sum of

the measures of the two arcs.

A

C

mABC = mAB + mBC

B

26.
Congruent arcs – two arcs of the same circle or of

congruent circles that have the same measure

congruent circles that have the same measure

27.
Arcs and Chords Theorem

In the same circle, or in congruent circles, two minor arcs are

congruent if and only if their corresponding chords are

congruent.

A

B

AB BC if and only if AB BC

C

In the same circle, or in congruent circles, two minor arcs are

congruent if and only if their corresponding chords are

congruent.

A

B

AB BC if and only if AB BC

C

28.
Perpendicular Diameter Theorem

If a diameter of a circle is perpendicular to a chord, then the

diameter bisects the chord and its arc.

F

DE EF, DG FG

E

G

D

If a diameter of a circle is perpendicular to a chord, then the

diameter bisects the chord and its arc.

F

DE EF, DG FG

E

G

D

29.
Perpendicular Diameter Converse

If one chord is a perpendicular bisector of another chord, then

the first chord is a diameter.

J

M

K

L

JK is a diameter of the circle.

If one chord is a perpendicular bisector of another chord, then

the first chord is a diameter.

J

M

K

L

JK is a diameter of the circle.

30.
Congruent Chords Theorem

In the same circle, or in congruent circles, two chords are

congruent if and only if they are equidistant from the center.

C

G

AB CD if and only if EF EG. E D

B

F

A

In the same circle, or in congruent circles, two chords are

congruent if and only if they are equidistant from the center.

C

G

AB CD if and only if EF EG. E D

B

F

A

31.
Example 1

Find the measure of each arc.

a. LM 70°

N L

P 70

b. MNL 360° - 70° = 290°

c. LMN 180° M

Find the measure of each arc.

a. LM 70°

N L

P 70

b. MNL 360° - 70° = 290°

c. LMN 180° M

32.
Example 2

Find the measures of the red arcs. Are the arcs congruent?

A

C

41

41

D

mAC = mDE = 41 E

Since the arcs are in the same circle, they are congruent!

Find the measures of the red arcs. Are the arcs congruent?

A

C

41

41

D

mAC = mDE = 41 E

Since the arcs are in the same circle, they are congruent!

33.
Example 3

Find the measures of the red arcs. Are the arcs congruent?

A

D

81

E

C

mDE = mAC = 81

However, since the arcs are not of the same circle or

congruent circles, they are NOT congruent!

Find the measures of the red arcs. Are the arcs congruent?

A

D

81

E

C

mDE = mAC = 81

However, since the arcs are not of the same circle or

congruent circles, they are NOT congruent!

34.
Example 4

B

Find mBC.

(3x + 11)

(2x + 48)

3x + 11 = 2x + 48

A

x = 37

D C

mBC = 2(37) + 48

mBC = 122

B

Find mBC.

(3x + 11)

(2x + 48)

3x + 11 = 2x + 48

A

x = 37

D C

mBC = 2(37) + 48

mBC = 122

35.
Inscribed angle – an angle whose vertex is on a circle

and whose sides contain chords of the circle

Intercepted arc – the arc that lies in the interior of an

inscribed angle and has endpoints on the angle

intercepted arc

inscribed angle

and whose sides contain chords of the circle

Intercepted arc – the arc that lies in the interior of an

inscribed angle and has endpoints on the angle

intercepted arc

inscribed angle

36.
Measure of an Inscribed Angle Theorem

If an angle is inscribed in a circle, then its measure is

half the measure of its intercepted arc.

A

1 C

mADB = mAB D

2 B

If an angle is inscribed in a circle, then its measure is

half the measure of its intercepted arc.

A

1 C

mADB = mAB D

2 B

37.
Example 1

Find the measure of the blue arc or angle.

E

a. S R b.

80

F

Q G

T

1

mQTS = 2(90 ) = 180 mEFG = (80 ) = 40

2

Find the measure of the blue arc or angle.

E

a. S R b.

80

F

Q G

T

1

mQTS = 2(90 ) = 180 mEFG = (80 ) = 40

2

38.
Congruent Inscribed Angles Theorem

If two inscribed angles of a circle intercept

the same arc, then the angles are

congruent.

A

B

C

D

C D

If two inscribed angles of a circle intercept

the same arc, then the angles are

congruent.

A

B

C

D

C D

39.
Example 2

It is given that mE = 75 . What is mF?

Since E and F both intercept D

the same arc, we know that the

angles must be congruent.

E

mF = 75

F

H

It is given that mE = 75 . What is mF?

Since E and F both intercept D

the same arc, we know that the

angles must be congruent.

E

mF = 75

F

H

40.
Inscribed polygon – a polygon whose vertices all lie on a

circle.

Circumscribed circle – A circle with an inscribed polygon.

The polygon is an inscribed polygon and

the circle is a circumscribed circle.

circle.

Circumscribed circle – A circle with an inscribed polygon.

The polygon is an inscribed polygon and

the circle is a circumscribed circle.

41.
Inscribed Right Triangle Theorem

If a right triangle is inscribed in a circle, then the hypotenuse

is a diameter of the circle. Conversely, if one side of an

inscribed triangle is a diameter of the circle, then the triangle

is a right triangle and the angle opposite the diameter is the

right angle.

A

B is a right angle if and only if AC

is a diameter of the circle. B

C

If a right triangle is inscribed in a circle, then the hypotenuse

is a diameter of the circle. Conversely, if one side of an

inscribed triangle is a diameter of the circle, then the triangle

is a right triangle and the angle opposite the diameter is the

right angle.

A

B is a right angle if and only if AC

is a diameter of the circle. B

C

42.
Inscribed Quadrilateral Theorem

A quadrilateral can be inscribed in a circle if and only if

its opposite angles are supplementary.

E

F

C

D

G

D, E, F, and G lie on some circle, C if and only if

mD + mF = 180 and mE + mG = 180 .

A quadrilateral can be inscribed in a circle if and only if

its opposite angles are supplementary.

E

F

C

D

G

D, E, F, and G lie on some circle, C if and only if

mD + mF = 180 and mE + mG = 180 .

43.
Example 3

Find the value of each variable.

D

a. b.

B z

G y 120 E

Q

A 2x

80

F

C

mD + mF = 180 mG + mE = 180

2x = 90

z + 80 = 180 y + 120 = 180

x = 45

z = 100 y = 60

Find the value of each variable.

D

a. b.

B z

G y 120 E

Q

A 2x

80

F

C

mD + mF = 180 mG + mE = 180

2x = 90

z + 80 = 180 y + 120 = 180

x = 45

z = 100 y = 60

44.
Tangent-Chord Theorem

If a tangent and a chord intersect at a point on a circle,

then the measure of each angle formed is one half the

measure of its intercepted arc.

B

1

m1 = mAB C

2

1 1

m2 = mBCA 2

2 A

If a tangent and a chord intersect at a point on a circle,

then the measure of each angle formed is one half the

measure of its intercepted arc.

B

1

m1 = mAB C

2

1 1

m2 = mBCA 2

2 A

45.
Example 1

Line m is tangent to the circle. Find mRST m

R

102

mRST = 2(102 )

S

mRST = 204

T

Line m is tangent to the circle. Find mRST m

R

102

mRST = 2(102 )

S

mRST = 204

T

46.
Try This!

Line m is tangent to the circle. Find m1

1 R

m1 = (150 )

2 1

m

m1 = 75

150

T

Line m is tangent to the circle. Find m1

1 R

m1 = (150 )

2 1

m

m1 = 75

150

T

47.
Example 2

BC is tangent to the circle. Find mCBD. C

A

(9x+20)

5x B

2(5x) = 9x + 20

10x = 9x + 20

x = 20

D

mCBD = 5(20 )

mCBD = 100

BC is tangent to the circle. Find mCBD. C

A

(9x+20)

5x B

2(5x) = 9x + 20

10x = 9x + 20

x = 20

D

mCBD = 5(20 )

mCBD = 100

48.
Interior Intersection Theorem

If two chords intersect in the interior of a circle, then the

measure of each angle is one half the sum of the

measures of the arcs intercepted by the angle and its

vertical angle.

1 D

m1 = (mCD + mAB) A

2

1

2

1

m2 = (mAD + mBC) C

2 B

If two chords intersect in the interior of a circle, then the

measure of each angle is one half the sum of the

measures of the arcs intercepted by the angle and its

vertical angle.

1 D

m1 = (mCD + mAB) A

2

1

2

1

m2 = (mAD + mBC) C

2 B

49.
Exterior Intersection Theorem

If a tangent and a secant, two tangents, or

two secants intersect in the exterior of a

circle, then the measure of the angle

formed is one half the difference of the

measures of the intercepted arcs.

If a tangent and a secant, two tangents, or

two secants intersect in the exterior of a

circle, then the measure of the angle

formed is one half the difference of the

measures of the intercepted arcs.

50.
Diagrams for Exterior

Intersection Theorem

B

A P

1

2

Q

C R

1

1 m2 = (mPQR - mPR)

m1 = (mBC - mAC) X 2

2

W

3

Z Y

1

m3 = (mXY - mWZ)

2

Intersection Theorem

B

A P

1

2

Q

C R

1

1 m2 = (mPQR - mPR)

m1 = (mBC - mAC) X 2

2

W

3

Z Y

1

m3 = (mXY - mWZ)

2

51.
Example 3

P

Find the value of x. 106

Q

1

x = (mPS + mRQ)

2 x

1 S

x = (106 +174 )

2 174 R

1

x= (280)

2

x = 140

P

Find the value of x. 106

Q

1

x = (mPS + mRQ)

2 x

1 S

x = (106 +174 )

2 174 R

1

x= (280)

2

x = 140

52.
Try This!

Find the value of x. T

40

1 S

x = (mST + mRU) x

2

U

1

x = (40 +120 )

2

R 120

1

x= (160)

2

x = 80

Find the value of x. T

40

1 S

x = (mST + mRU) x

2

U

1

x = (40 +120 )

2

R 120

1

x= (160)

2

x = 80

53.
Example 4

Find the value of x.

1

72 = (200 - x ) 200

2

144 = 200 - x

x 72

x = 56

Find the value of x.

1

72 = (200 - x ) 200

2

144 = 200 - x

x 72

x = 56

54.
Example 5

Find the value of x.

A

mABC = 360 - 92 B

mABC = 268 92 x

1

x= (268 - 92) C

2

1

x = (176)

2

x = 88

Find the value of x.

A

mABC = 360 - 92 B

mABC = 268 92 x

1

x= (268 - 92) C

2

1

x = (176)

2

x = 88